Reduction of Infinite Dimensional Systems to Finite Dimensions: Compact Convergence Approach

Effective particle methods for Fisher–Kolmogorov equations: Theory and applications to brain tumor dynamics

“…Finally, we can also obtain the analytical expression for the velocity v(t) by combining (14b) and (16) to get…”

“…In that context there has been a great interest on different types of finite-dimensional descriptions of the dynamics of systems ruled by PDEs in a variety of contexts (see e.g. [13,14,15,16] and references therein). However a simple procedure such as the one provided by effective particle methods that allows applied scientists to reduce the dynamics of a partial differential equation with solitary waves to a set of finite dimensional simple equations for the solitary wave parameters is not available yet.…”

Rate of convergence of attractors for singularly perturbed semilinear problems

“…In this section we consider a more general class of linear operators than that seen in the Section 2. We will observe that for this class of operators, that was considered in [9], the theory developed in the previous section can be applied with some additional a priori estimates. Definition 7.1.…”

“…There are many parabolic problems whose asymptotic behavior is dictated by a system of Morse-Smale ordinary differential equations, for example, reaction diffusion equation where the diffusion coefficient become very large in all domain and reaction diffusion equation where the diffusion coefficient is very large except in a neighborhood of a finite number of points where it becomes small. These kind of problems was considered in the works [9,11,12,13,14] and [16], where well-posedness, functional setting and convergence of attractors was studied. In general it is considered a family of parabolic problems depending on a positive parameter ε and when ε converges to zero, it is obtained a limiting ordinary differential equation that contains all dynamic of the problem.…”

“…Lemma 9.2. For g ∈ L 2 with g L 2 ≤ 1 and ε ∈ (0, ε 0 ], let u ε be the solution of elliptic problem A ε u ε = g and let u 0 = (u 0 1 , u 0 2 ) be the solution of A 0 u 0 = M ε g. Then there is a constant C > 0, independent of ε, such that u ε − E ε u 0 2 X 1 2 ε ≤ C(|a 1 − a 1 | + |l 1 − l 1 | + ε Proof: The arguments used here was inspired in [9].…”